A chord-stretching map of a convex loop is an isometry

Abstract

Let R" be n-dimensional Euclidean space, and let c~: [0, L] -~ R" and fl: [0, L] -, ~" be closed rectifiable arcs in N" of the same total length L which are parametrized via their arc length. fl is said to be a chord-stretched version of • if for each 0 ~< s ~< t ~< L, Is(t)-e(s)l ~< Ifl(t) -fl(s)l, fl is said to be convex if fl is simple and if fl([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if ~ were simple then there existed a convex chord-stretched version fl of e. This result led Professor Yang Lu to conjecture that if ct were convex and fl were a chord-stretched version of ct then ~ and fl would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where c~ and ~ are C 2 curves. In this paper we prove the conjecture in general.

Total Curvature and the Isoperimetric Inequality in Cartan–Hadamard Manifolds

“…We say that γ majorizes γ provided that there exists a nonexpanding map f : C → X with f • γ = γ. The curve γ has also been called an "unfolding" [15,25] or "chord-stretching" [40,46] of γ. We call f the majorization map.…”

Convexification of polygons by length preserving transformations

“…See also [6,8] and the references therein. More results on edge-length preserving transformations and chord stretching can be found in [4,12,13,14].…”

“…Observe that for the operation to be well-defined we need that pi−1 and pi+1 are distinct. This operation belongs to the larger class of edge-length preserving transformations, when applied to polygons [4,12,13,14]. It seems to have been used for the first time by Millet.…”

On convexification of polygons by pops

“…Observe that for the operation to be well-defined we need that p i−1 and p i+1 are distinct. This operation belongs to the larger class of edge-length preserving transformations, when applied to polygons [5,19,20,21,22]. It seems to have been used for the first time by Millet [16].…”

“…There is an extensive bibliography pertaining to these subjects [1,2,3,4,5,6,8,9,10,12,13,14,15,17,18,23,24,25]. See also [5,19,20,21,22] for more results on edge-length preserving transformations and chord stretching.…”